Power variations and limit theorems for stochastic processes controlled by fractional Brownian motions
In the setting of rough paths, let \((y_t)_{t\in[0,1]}\) be a stochastic process controlled by a fractional Brownian motion with Hurst parameter \(H\leq1/2\). The corresponding power variation of order \(p>0\) is\N\[\N\sum_{k=0}^{n-1}|y_{t_{k+1}}-y_{t_k}|^p\,,\N\]\Nwhere \(0=t_0<t_1<\cdots<t_n=1\). The authors establish limit theorems for these power variations, distinguishing the three cases \(H<1/4\), \(H=1/4\) and \(H>1/4\). The limit theorems are based on a decomposition of the power variation into weighted random sums and remainder terms. When \(H<1/4\) the authors show convergence in probability of the centred power variation at rate \(n^{-2H}\) as \(n\to\infty\). When \(H\geq1/4\) there is convergence in law at rate \(n^{-1/2}\).
scientific article; zbMATH DE number 7922283
scientific article